Proof of the derivative of an elementary function is also elementary

Question

I heard of the result that the derivative of an elementary function is also elementary long ago. Now I want to prove it rigorously. I found this answer(I didn't comment because it was an old post):https://math.stackexchange.com/q/2195559. It actually proved that: (1). the derivative of $\exp,\log,\operatorname{id}$ and constant is elementary. (2). If we assume that f and g have elementary derivatives, then $f+g,f-g,fg,f/g,f\circ g$ also have elementary derivatives. So far so good. But what confuses me is how does this 2 facts imply the original proposition(the derivative of any elementary function is also elementary)? I mean isn't the fact (2) seems too weak?
I think the keypoint lies in the construction of the elementary functions. I use the definition of the elementary functions to be:

1. $\exp,\log,\operatorname{id}$ and constant is elementary.
2. the sum, difference, product, quotient, composition of 2 elementary functions is elementary.
   
   Need help!

Answer 1

The closure under differentiation can be proven by induction.

The result is clear for « base functions » ($\ln x, e^x)...$

If a map is a sum, the product, the composition... of two simple functions, then it follows that its derivative is a simple function based on the induction hypothesis and the differentiation rules of the sum, the product, the composition... of two functions.

As any simple function can be obtained by induction using previous rules according to the very definition of a simple function, we’re done.

Answer 2

Let me try to organize what you have a little more neatly, maybe this will clear things up.

First, your definition of elementary function:

$ \bullet \exp, \log, \text{id}, \text{constant}\\ \bullet \text{The sum of two elementary functions, i.e. }f+g \\ \bullet \text{The difference of two elementary functions, i.e. }f-g \\ \bullet \text{The product of two elementary functions, i.e. }fg \\ \bullet \text{The quotient of two elementary functions, i.e. }\frac{f}{g} \\ \bullet \text{The composition of two elementary functions, i.e. }f \circ g $

Now, the proof you posted proves first that $\exp, \log$, id and constant functions have elementary derivatives, so the claim holds for the first bullet point in the list above. Then it goes on to prove that it holds for sums (2nd bullet point), differences (3rd bullet), products (4th bullet), quotients (5th bullet) and compositions (6th bullet).

So it really has shown that all the functions that fall under the definition of elementary function have elementary derivatives! And as mathcounterexamples.net pointed out in his answer, this can be expanded to the sum, product, etc. of more than two functions by induction.